Once you find your missing AirTag, you can stop playing the sound by tapping Stop Sound.
Then unlock your iPhone, hold the AirTag near, and wait for the pairing dialog to pop up on the screen.
You do not have to charge an AirTag, but you will have to change its battery.
I was told that Apple was not going to be looking at this issue as a systemic issue until they receive enough calls about it.
With all that out of the way, I tracked my family with my AirTag and Tile Pro in a variety of scenarios.